Invariant bilinear forms under the rigid motions of a regular polygon

TL;DR

This paper calculates the number of degree n representations of dihedral groups and analyzes invariant bilinear forms, including their existence and non-degeneracy, with implications for physical sciences.

Contribution

It provides explicit computations of invariant bilinear forms for dihedral groups and discusses their non-degeneracy, advancing understanding in symmetry representations.

Findings

Number of n-degree representations for dihedral groups computed

Dimensions of invariant bilinear form spaces determined

Conditions for existence of non-degenerate invariant forms analyzed

Abstract

In this paper, for n a positve integer, we compute the number of n degree representations for a dihedral group G of order 2m, m \geq 3 and the dimensions of the corresponding spaces of G invariant bilinear forms over a complex field C. We explicitly discuss about the existence of a non-degenerate invariant bilinear form. The results are important due to their applications in the studies of physical sciences.